A168364 -id:A168364 - cp's OEIS frontend

A229870 T(n,k)=Number of n X n 0..k arrays with corresponding row and column sums equal.

2, 3, 8, 4, 27, 80, 5, 64, 1215, 2432, 6, 125, 8704, 384183, 247552, 7, 216, 40625, 15106048, 923742873, 88060928, 8, 343, 143856, 266515625, 354003288064, 17451302074317, 112371410944, 9, 512, 420175, 2805425280, 36821326171875

Offset: 1

R. H. Hardin, Oct 01 2013

Table starts
......2.........3............4..............5................6...........7
......8........27...........64............125..............216.........343
.....80......1215.........8704..........40625...........143856......420175
...2432....384183.....15106048......266515625.......2805425280.20610104767
.247552.923742873.354003288064.36821326171875.1656812779036416

			Some solutions for n=4 k=4
..0..0..0..1....0..0..0..1....0..0..0..0....0..0..1..1....0..0..1..1
..0..1..2..1....0..0..3..4....0..0..3..3....0..1..0..3....0..0..2..2
..1..0..0..3....1..4..2..0....0..4..0..2....1..3..4..0....1..3..4..1
..0..3..2..1....0..3..2..2....0..2..3..2....1..0..3..2....1..1..2..0

R. H. Hardin, Table of n, a(n) for n = 1..43

Row 2 is A000578(n+1)

Row 3 is A168364(n+1)

Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^3 + 3*n^2 + 3*n + 1
n=3: [polynomial of degree 7]
n=4: [polynomial of degree 13]

A059977 a(n) = binomial(n+2, 2)^4.

Original entry on oeis.org

1, 81, 1296, 10000, 50625, 194481, 614656, 1679616, 4100625, 9150625, 18974736, 37015056, 68574961, 121550625, 207360000, 342102016, 547981281, 855036081, 1303210000, 1944810000, 2847396321, 4097152081, 5802782976, 8100000000, 11156640625, 15178486401

Offset: 0

Robert G. Wilson v, Mar 06 2001

nonn

Number of 4-dimensional cage assemblies.

See Chap. 61, "Hyperspace Prisons", of C. Pickover's book "Wonders of Numbers" for full explanation of "cage numbers."

			1 = (1 + 1)/2, 81 = (33 + 129)/2, 1296 = (276 + 2316)/2, 10000 = (1300 + 18700)/2, ... - _Philippe Deléham_, May 25 2015

Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Oxford University Press, 2001, p. 325.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000217, A000539, A000541, A059827, A059860.

Cf. A168364 (first differences).

with (combinat):seq(mul(stirling2(n+1,n),k=1..4),n=1..24); # Zerinvary Lajos, Dec 16 2007

m = 4; Table[ ( (n^m)(n + 1)^m )/(2^m), {n, 1, 30} ]

a(n) = { ((n + 1)*(n + 2)/2)^4 } \\ Harry J. Smith, Jun 30 2009

[stirling_number2(n+1,n)^4for n in range(1,25)] # Zerinvary Lajos, Mar 14 2009

L(n) = ((n^m)(n + 1)^m)/(2^m) where m is the dimension, which in this case is 4.
O.g.f.: -(1+72*x+603*x^2+1168*x^3+603*x^4+72*x^5+x^6)/(-1+x)^9. - R. J. Mathar, Mar 31 2008
a(n) = A000217(n+1)^4. - R. J. Mathar, Dec 13 2011
a(n) = (A000539(n+1) + A000541(n+1))/2. - Philippe Deléham, May 25 2015
From Amiram Eldar, May 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 160*Pi^2/3 + 16*Pi^4/45 - 560.
Sum_{n>=0} (-1)^n/a(n) = 560 - 640*log(2) - 96*zeta(3). (End)

Better definition from Zerinvary Lajos, May 23 2006

A168351 a(n) = n^5*(n+1)/2.

Original entry on oeis.org

0, 1, 48, 486, 2560, 9375, 27216, 67228, 147456, 295245, 550000, 966306, 1617408, 2599051, 4033680, 6075000, 8912896, 12778713, 17950896, 24760990, 33600000, 44925111, 59266768, 77236116, 99532800, 126953125, 160398576, 200884698

Offset: 0

N. J. A. Sloane, Dec 11 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Sequences of the form n^5*(n^k + 1)/2: A000584 (k=0), this sequence (k=1), A168364 (k=2), A168371 (k=3), A168372 (k=4), A071236 (k=5), A168412 (k=6), A168432 (k=7), A168462 (k=8), A168471 (k=9), A168507 (k=10).

[n^5*(n+1)/2: n in [0..30]]; // Vincenzo Librandi, Aug 28 2011

Table[n^5*(n + 1)/2, {n, 0, 40}] (* Wesley Ivan Hurt, Aug 13 2015 *)

def A168351(n): return n^4*binomial(n+1,2)
print([A168351(n) for n in range(41)]) # G. C. Greubel, Mar 20 2025

a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} Sum_{m=1..n} (i+j+k-l-m). - Wesley Ivan Hurt, Aug 13 2015
From G. C. Greubel, Mar 20 2025: (Start)
G.f.: x*(1 + 41*x + 171*x^2 + 131*x^3 + 16*x^4)/(1-x)^7.
E.g.f.: (1/2)*x*(2 + 46*x + 115*x^2 + 75*x^3 + 16*x^4 + x^5)*exp(x). (End)

A357178 First differences of cubes of triangular numbers.

Original entry on oeis.org

0, 1, 26, 189, 784, 2375, 5886, 12691, 24704, 44469, 75250, 121121, 187056, 279019, 404054, 570375, 787456, 1066121, 1418634, 1858789, 2402000, 3065391, 3867886, 4830299, 5975424, 7328125, 8915426, 10766601, 12913264, 15389459, 18231750, 21479311, 25174016, 29360529

Offset: 0

Kelvin Voskuijl, Sep 16 2022

Row sums of centered hexagonal numbers A003215 treated as a regular triangle.

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A059827 (cubes of triangular numbers).
Cf. A000578 (for squares) and A168364 (for fourth powers) of triangular numbers.
Cf. A000217 (triangular numbers), A003215.

a[n_] := (n^3 + 3*n^5)/4; Array[a, 35, 0] (* Amiram Eldar, Sep 18 2022 *)

a(n) = n^3*(3*n^2+1)/4 \\ Charles R Greathouse IV, Sep 19 2022

a(n) = (n^3 + 3*n^5)/4.

G.f.: x*(1 + 20*x + 48*x^2 + 20*x^3 + x^4)/(1 - x)^6. - Stefano Spezia, Sep 19 2022

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A229870 T(n,k)=Number of n X n 0..k arrays with corresponding row and column sums equal.

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A059977 a(n) = binomial(n+2, 2)^4.

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A168351 a(n) = n^5*(n+1)/2.

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A357178 First differences of cubes of triangular numbers.

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